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A103519 a(1) = 1, a(n) = sum of n successive numbers starting with a(n-1) + 1. 2
1, 5, 21, 94, 485, 2931, 20545, 164396, 1479609, 14796145, 162757661, 1953092010, 25390196221, 355462747199, 5331941208105, 85311059329816, 1450288008607025, 26105184154926621, 495998498943605989, 9919969978872119990 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Eigensequence of a triangle with the natural numbers (1, 2, 3,...) as the right border, the triangular series (1, 3, 6,...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..449

FORMULA

a(n+1) = k(k+1)/2 - a(n)*(a(n)+1)/2, where k = a(n) + n +1.

a(n) = Sum[i=0..n] n!/(n-i)! * (n-i)(n-i+1)/2 = Sum[i=0..n] n!/(n-i)! * A000217(n-i). For n>2, a(n) = 3*n*(n-1)/2*floor((n-2)!*e)+n, where e=exp(1). - Max Alekseyev, Feb 14 2005

a(n) = n*a(n-1) + n(n+1)/2. - Emeric Deutsch, Mar 16 2008

a(n) ~ 3*sqrt(Pi/2)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016

EXAMPLE

a(2) = 2+3 = 5, a(3) = 6+7+8 = 21, a(4) = 22 +23 +24 +25.

MAPLE

a[1]:=1: for n from 2 to 20 do a[n]:=n*a[n-1]+(1/2)*n*(n+1) end do: seq(a[n], n=1..20); # Emeric Deutsch, Mar 16 2008

MATHEMATICA

RecurrenceTable[{a[1]==1, a[n]==n*a[n-1]+(n(n+1))/2}, a, {n, 20}] (* Harvey P. Dale, Nov 05 2013 *)

PROG

(PARI) { t(n) = n*(n+1)/2 } { a(n) = sum(i=0, n, n!/(n-i)!*t(n-i)) } { a2(n) = 3*t(n-1)*floor((n-2)!*exp(1))+n } \\ Max Alekseyev, Feb 14 2005

CROSSREFS

Cf. A103520.

Sequence in context: A116904 A126952 A273570 * A178876 A202513 A343349

Adjacent sequences:  A103516 A103517 A103518 * A103520 A103521 A103522

KEYWORD

easy,nonn

AUTHOR

Amarnath Murthy, Feb 10 2005

EXTENSIONS

More terms from Max Alekseyev, Feb 14 2005

STATUS

approved

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Last modified October 20 18:00 EDT 2021. Contains 348111 sequences. (Running on oeis4.)